Real Numbers

Real Numbers

1. Euclid's Division Lemma

   • any Positive integer 'a' can be divided by positive integer 'b' such that remainder 'r'< 'b' is found. [theorem 1.1]
     • Simply Long Division Method
       • Algebraically: a=bq+r, 0 ≤r≥ b
   • Application: computing HCF of 2 positive integers
     • to obtain HCF: c,d are the numbers,c+dq+r,0 ≤r≥ b,if 'r'=0 then 'd' is the HCf else continue the process with 'd' and 'r' until remainder=0,devisor in that stage is the HCF

2. Fundamental Theorem of Arithmetic

   • Every Composite Number can be Expressed as a Product of Primes in a Unique way. [theorem 1.2]
     • Application
       • Proving Irrationality of Numbers
         • When 'p' is a Prime no. and 'p' divides 'a²' then 'p' also divides 'a'. [theorem 1.3]
       • Finding Exact Decimal Expansion of a Rational Number
         • if 'x' is Rational No. Whose decimal Expansion is Terminating,Then it can be Expressed as 'p/q', where "p" and 'q' are co primes and factors of 'q' is in form 2^n*5^m,and 'n'&'m' are non-negative integers. [theorem 1.5]
           • If prime factorizing 'q' gives non 2^n*5^m form then the Decimal Expansion is non-terminating and repeating. [theorem 1.7]
       • HCF and LCM of two numbers by Prime Factorization
     • Here 'expressed' means factorized and 'unique' doesn't include the order of factors